http://www.jsoftware.com/pipermail/general/2005-December/026024.html
----- Original Message ----- From: Tracy Harms <[EMAIL PROTECTED]> Date: Wednesday, May 28, 2008 12:15 Subject: [Jchat] Iverson notations and infinite sets To: [email protected] > John Randall wrote, in the Programming forum: > > > I think the author of the Wikipedia article is trying to get > at this, > > so you would accept 1+1=2 (mod 12) but not 9+4=1 (mod > 12). In my > > opinion, this fixates on the distinguished representatives rather > > than the equivalence classes, and if you do that, you will go wrong > > somewhere else. > > Thank you for this assessment of that flawed Wikipedia paragraph. > > Because equivalence classes are infinite sets, my curiosity is piqued: > How should Iverson notation be applied when dealing with > infinite sets? > > It seems that for this use we must move to statements that would > not be > valid under the requirements of executability that J and APL entail. > Yet, it would be nice to be able to refer to things like these > equivalence classes, integers, rational numbers, real numbers, > and the > complex plane within a near-J notational structure. > > I've put this to the Chat forum because it departs from J > proper, but if > the moderators think it is a better fit for the Programming > forum we can > move it back there. > > Another area where I'm not entirely sure how to apply J are statements > of formal logic such as "for all" and "there exists" (inverted > capitalsA and E, respectively.) > > In general, I wish to explore a return to the roots of J. The > abstraction of tacit notation is itself a triumph along those > lines; the > way it removes specification of particulars allows us to refer > strictlyto the functions. The arguments could in many cases be > infinite sets, > setting aside the needs of implementation. The limit power > (^:_) also > provides something along these lines. > > As an example of where my curiosity has wandered, I've not been > able to > decide whether (i: _j_) naturally denotes the Rationals, or only a > subset of them. (If "_" were taken as a value, which it must not > be, it > would denote the Integers.) ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
